Trans-dimensional inverse problems, model comparison and the evidence

被引:267
作者
Sambridge, M. [1 ]
Gallagher, K.
Jackson, A.
Rickwood, P.
机构
[1] Australian Natl Univ, Res Sch Earth Sci, Canberra, ACT 0200, Australia
[2] Univ London Imperial Coll Sci Technol & Med, Dept Earth Sci & Engn, London SW7 2BP, England
[3] ETH, Inst Geophys, CH-8093 Zurich, Switzerland
关键词
evidence; inverse problems; model comparison; parametrization;
D O I
10.1111/j.1365-246X.2006.03155.x
中图分类号
P3 [地球物理学]; P59 [地球化学];
学科分类号
0708 ; 070902 ;
摘要
In most geophysical inverse problems the properties of interest are parametrized using a fixed number of unknowns. In some cases arguments can be used to bound the maximum number of parameters that need to be considered. In others the number of unknowns is set at some arbitrary value and regularization is used to encourage simple, non-extravagant models. In recent times variable or self-adaptive parametrizations have gained in popularity. Rarely, however, is the number of unknowns itself directly treated as an unknown. This situation leads to a trans-dimensional inverse problem, that is, one where the dimension of the parameter space is a variable to be solved for. This paper discusses trans-dimensional inverse problems from the Bayesian viewpoint. A particular type of Markov chain Monte Carlo (MCMC) sampling algorithm is highlighted which allows probabilistic sampling in variable dimension spaces. A quantity termed the evidence or marginal likelihood plays a key role in this type of problem. It is shown that once evidence calculations are performed, the results of complex variable dimension sampling algorithms can be replicated with simple and more familiar fixed dimensional MCMC sampling techniques. Numerical examples are used to illustrate the main points. The evidence can be difficult to calculate, especially in high-dimensional non-linear inverse problems. Nevertheless some general strategies are discussed and analytical expressions given for certain linear problems.
引用
收藏
页码:528 / 542
页数:15
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